## Franklin's Notes

### Galois connection

A Galois connection between two preorders $P$ and $Q$ consists of a pair of functions $f:P\to Q$ and $g:Q\to P$ such that We say that $f$ is the left adjoint and $g$ is the right adjoint of the Galois connection. This terminology is (deliberately!) borrowed from the definition of adjoint functors in category theory , for if $P$ and $Q$ are regarded as abstract categories and $f,g$ as functors between them, this definition is equivalent to $f,g$ comprising an adjunction.

A notable fact about Galois connections is that they preserve meets and joins, when they exist. Conversely, monotone maps that preserve existent meets and joins are right or left adjoints (of some other monotone map), respectively. This is closely related to the preservation of limits and colimits by adjoint functors.